If one wanted to collaterally estop man from relitigating an issue previously decided in Round 1, would one look to the findings in the quasi-judicial grievance proceeding, the trial court order, the appellate court decision, or some combination of all three?

It looks like he had only one potentially "full and fair hearing on the merits" as to the grievance itself. Most jurisdictions require this for collateral estoppel. That was at the quasi-judicial hearing.

Of course, if you want to estop him as to the issue of whether the trial court properly denied the writ, then, if he had a full and fair hearing on that issue in the trial court (and the appellate court), he can invoke CE as to that issue only.

But in the petition before the trial court, all the evidence that was available in the grievance is also reviewed by the TC judge in a limited trial de novo. Say that he makes his own findings as to what happened (on a particular issue the hearing officer did not reach), which are different, but not inconsistent with what the quasi-judicial hearing officer held. Is a petition for writ of mandate not a "full and fair hearing on the merits"?

But in the petition before the trial court, all the evidence that was available in the grievance is also reviewed by the TC judge in a limited trial de novo. Say that he makes his own findings as to what happened (on a particular issue the hearing officer did not reach), which are different, but not inconsistent with what the quasi-judicial hearing officer held. Is a petition for writ of mandate not a "full and fair hearing on the merits"?

"Limited trial de novo" hardly equates with "full and fair hearing" to me. In addition, CE usually requires that the issue to be estopped is identical with the issue previously litigated. A writ of mandate (mandamus) is an extraordinary writ. At the trial court level, the court was facing the question of whether or not to compel an official (the quasi-judicial officer) to do something. So, while the evidence may have been viewed de novo, the review was done only with regard to how bad the officer screwed up. The issues are therefore not identical. In the first case the issue was, "Is the quasi-judicial finding so contrary to law that we should compel that officer to change his ruling." The issue that the man will now try to litigate is "Was I wrongfully terminated?"

Well yeah, the issues aren't identical, but in determining whether or not to compel the officer to do something, the trial court had to decide whether the hearing officer's findings were consistent with the evidence. In reviewing the evidence and exercising its independent judgment, the court could find either the guy was or wasn't wrongfully terminated.

So say the TC finds that the guy was not wrongfully terminated and denies the writ. The findings of fact/law in this ruling cannot be a basis for collaterally estopping the new suit?

The quasi-judicial ruling, combined with the trial court's judgment effectively sustaining that ruling, constitutes a full and fair opportunity to be heard on the issue of wrongful termination, and the trial court's ruling constitutes a final judgment on that issue.

If I had to argue against CE:

There is no court judgment on the issue of wrongful termination. There is only a judgment on the mandamus issue, whereby the plaintiff was not fully heard on the merits of the case, but rather the trial judge reviewed the record in its entirety only to determine whether there was extraordinary abuse of discretion by the quasi-judicial officer.